What is a rational expression?

A rational expression is a fraction where both the numerator and denominator are polynomials. For example, (x² − 4)/(x − 2) is a rational expression. The key requirement is that the denominator cannot equal zero. Rational expressions are to polynomials what rational numbers (fractions) are to integers.

How do you simplify a rational expression?

To simplify a rational expression: (1) Factor both the numerator and denominator completely. (2) Identify and cancel any common factors that appear in both. (3) Write the remaining factors as the simplified expression. For example, (x² − 4)/(x² − 4x + 4) factors to (x−2)(x+2)/(x−2)², which simplifies to (x+2)/(x−2). Always note the domain restrictions from the cancelled factors.

How do you multiply rational expressions?

To multiply rational expressions: (1) Factor all numerators and denominators completely. (2) Cancel any common factors between any numerator and any denominator (cross-cancelling). (3) Multiply the remaining numerators together and the remaining denominators together. For example, (x+2)/(x−1) × (x−1)/(x+3) — cancel the (x−1) factor — gives (x+2)/(x+3).

How do you divide rational expressions?

To divide rational expressions, multiply the first expression by the reciprocal (flip) of the second. That is, (a/b) ÷ (c/d) = (a/b) × (d/c). Then simplify as you would any multiplication problem. For example, (x²−9)/(x+2) ÷ (x−3)/1 = (x²−9)/(x+2) × 1/(x−3) = (x−3)(x+3)/((x+2)(x−3)) = (x+3)/(x+2).

What is the LCD of rational expressions?

The LCD (Least Common Denominator) of rational expressions is the simplest polynomial that each denominator divides into evenly — the polynomial LCM. To find it: factor each denominator completely, then take each distinct factor raised to the highest power it appears in any denominator. For example, the LCD of 1/(x−1) and 1/(x²−1) is (x−1)(x+1) since x²−1 = (x−1)(x+1).

Why can't you cancel terms (only factors)?

Cancellation works by dividing, and you can only divide out factors (things multiplied together), not terms (things added together). For example, in (x+3)/x, you cannot cancel the x because x+3 is a sum, not a product. Writing (x+3)/x = 3 is a common error. The correct simplification is 1 + 3/x. Similarly, (x²+x)/x = x(x+1)/x = x+1 is valid because x is a factor of the numerator.

Can this calculator handle higher-degree polynomials?

Yes, this calculator supports polynomials of any degree with integer coefficients. It uses the rational root theorem to find roots and factor polynomials. For polynomials with no rational roots (like x² + 1), the polynomial is treated as irreducible and left unfactored. The calculator works best with polynomials up to degree 4-5 that have integer or simple rational roots.

Rational Expressions Calculator

Expression

Numerator

Denominator

Try an example

Use ^ for exponents: x^2 means x². Supports polynomials in x with integer coefficients.

Simplified Expression

x + 2

x − 2

Domain Restrictions

Values of x where the original expression is undefined

x ≠ 2

Step-by-Step Solution

7 steps to reach the result

Original expression

(x² − 4) / (x² − 4x + 4)

Factor numerator

(x − 2)(x + 2)

Factor denominator

(x − 2)²

Common factor (GCD)

x − 2

Cancel common factors

(x + 2) / (x − 2)

Result

(x + 2) / (x − 2)

Domain restrictions

x ≠ 2

What Are Rational Expressions?

The building blocks of algebraic fractions

A rational expression is a fraction where both the numerator and denominator are polynomials. Just as rational numbers are ratios of integers (like 3/4), rational expressions are ratios of polynomials (like (x² − 4)/(x − 2)).

P(x)/Q(x)

Ratio of polynomials

Q(x) ≠ 0

Denominator not zero

Factor & cancel

Simplification method

Key idea: Simplifying rational expressions works exactly like simplifying numeric fractions — factor both parts, then cancel common factors. The difference is that you factor polynomials instead of numbers.

How to Simplify Rational Expressions

The 3-step factoring method

Step 1.Factor the numerator and denominator

Break each polynomial into its irreducible factors. Use techniques like GCF extraction, difference of squares (a² − b² = (a−b)(a+b)), perfect square trinomials, and the AC method for quadratics.

Step 2.Cancel common factors

Any factor that appears in both numerator and denominator can be divided out. Each cancelled factor creates a domain restriction (a "hole" in the graph).

Step 3.State the domain restrictions

The simplified expression equals the original for all x except where the original denominator was zero. Always list these excluded values.

Worked Example

Simplify (x² − 4) / (x² − 4x + 4)

= (x − 2)(x + 2) / (x − 2)²

= (x + 2) / (x − 2), x ≠ 2

Operations on Rational Expressions

Adding, subtracting, multiplying, and dividing

Operation	Formula
Addition	a/b + c/d = (ad + bc) / bd
Subtraction	a/b − c/d = (ad − bc) / bd
Multiplication	(a/b) × (c/d) = ac / bd
Division	(a/b) ÷ (c/d) = ad / bc

Add / Subtract

Find LCD, rewrite each fraction, combine numerators, simplify

Multiply / Divide

Cross-cancel first, then multiply straight across (flip for division)

Domain Restrictions Explained

Understanding holes and vertical asymptotes

A rational expression is undefined wherever its denominator equals zero. These excluded values form the domain restrictions and come in two types:

Removable (Hole)

Factor cancels from numerator and denominator. The graph has a "hole" at that x-value.

(x−2)(x+1) / (x−2)

= x + 1, hole at x = 2

Non-removable (Asymptote)

Factor remains in the denominator after simplification. The graph has a vertical asymptote.

(x+1) / (x−3)

= asymptote at x = 3

Important: When you simplify a rational expression, the cancelled factors still restrict the domain. Always state domain restrictions based on the original denominator, not the simplified one.

Key Factoring Techniques

Essential methods for working with rational expressions

GCF (Greatest Common Factor)

6x² + 9x = 3x(2x + 3)

Difference of Squares

x² − 9 = (x − 3)(x + 3)

Perfect Square Trinomial

x² + 6x + 9 = (x + 3)²

Trinomial (AC Method)

2x² + 5x + 3 = (2x + 3)(x + 1)

Sum/Difference of Cubes

x³ − 8 = (x − 2)(x² + 2x + 4)

Finding the LCD (Least Common Denominator)

Essential for adding and subtracting rational expressions

To add or subtract rational expressions, you need a common denominator — specifically, the least common denominator (LCD). The LCD is the LCM (least common multiple) of the denominators.

Step 1.Factor each denominator completely

x² − 1 = (x−1)(x+1), x − 1 = (x−1)

Step 2.Include each factor at its highest power

LCD = (x−1)(x+1)

Step 3.Multiply each fraction by the missing factors

Second fraction gets multiplied by (x+1)/(x+1)

Common Mistakes to Avoid

Frequent errors when working with rational expressions

Do This

Factor completely before cancelling any terms
Cancel only factors, never individual terms
State domain restrictions from the original expression
Find the LCD before adding or subtracting
Flip the second fraction when dividing, then multiply

Avoid This

Cancelling terms instead of factors: (x+3)/x ≠ 3
Forgetting domain restrictions after simplifying
Adding fractions without a common denominator
Distributing the denominator: (a+b)/c ≠ a/(c) + b
Confusing (x²−4)/(x−2) with x²−4/(x−2)

Where Rational Expressions Are Used

Practical applications beyond algebra class

Physics & Engineering

Ohm's law for parallel resistors (1/R = 1/R₁ + 1/R₂), lens equations, and transfer functions in control systems all involve adding and simplifying rational expressions.

Rate & Work Problems

When two workers complete a job at different rates, their combined rate is the sum of rational expressions: 1/t₁ + 1/t₂ = 1/t_total.

Chemistry & Medicine

Dilution formulas (C₁V₁/V₂), drug concentration models over time, and reaction rate equations frequently use rational expressions.

Economics & Finance

Average cost functions, supply-demand equilibrium models, and elasticity calculations are expressed as ratios of polynomials.

Frequently Asked Questions

Common questions about simplifying and operating on rational expressions

Domain restrictions are values of the variable that make the denominator equal to zero. These values are excluded from the domain because division by zero is undefined. For example, in (x+1)/(x−3), the restriction is x ≠ 3. When you simplify and cancel factors, the cancelled values are still domain restrictions — they create "holes" in the graph rather than vertical asymptotes, but the expression is still undefined there.

To add or subtract rational expressions: (1) Factor each denominator. (2) Find the Least Common Denominator (LCD) by including each factor at its highest power. (3) Rewrite each fraction with the LCD as denominator by multiplying by the appropriate form of 1. (4) Add or subtract the numerators. (5) Simplify the result by factoring and cancelling common factors. For example, 1/x + 1/(x+1) = (x+1)/(x(x+1)) + x/(x(x+1)) = (2x+1)/(x(x+1)).

Both are points where a rational expression is undefined, but they differ graphically. A hole (removable discontinuity) occurs when a factor cancels from both numerator and denominator — the graph has a single missing point. A vertical asymptote (non-removable discontinuity) occurs when a factor remains in the denominator after simplification — the graph shoots toward infinity. For example, in (x−2)(x+1)/(x−2), x=2 is a hole. In (x+1)/(x−3), x=3 is a vertical asymptote.

This calculator uses exact integer arithmetic for polynomial operations, so results are mathematically exact for inputs with integer coefficients. The factoring uses the rational root theorem, which finds all rational roots. Polynomials that are irreducible over the rationals (like x² + 1) will not be factored further, which is mathematically correct. Always verify critical homework or exam answers by substituting test values into both the original and simplified expressions.

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Last updated Apr 15, 2026